Denseness results in the theory of algebraic fields

Anscombe, Sylvy; Dittmann, Philip; Fehm, Arno

doi:10.1016/j.apal.2021.102973

Cited by 2 publications

(2 citation statements)

References 10 publications

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“…show in Corollary 6.5 that for every open-closed subset of the p-adic spectrum of F , the associated holomorphy ring is diophantine. A further application can be found in the forthcoming work [ADF18], in which we use the results of this paper to show that rings of formal power series over number fields are Z-diophantine in their quotient fields.…”

Section: Introductionmentioning

confidence: 96%

A p-adic analogue of Siegel's Theorem on sums of squares

Anscombe,

Dittmann,

Fehm

2019

Preprint

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Siegel proved that every totally positive element of a number field K is the sum of four squares, so in particular the Pythagoras number is uniformly bounded across number fields. The p-adic Kochen operator provides a p-adic analogue of squaring, and a certain localisation of the ring generated by this operator consists of precisely the totally p-integral elements of K. We use this to formulate and prove a p-adic analogue of Siegel's theorem, by introducing the p-Pythagoras number of a general field, and showing that this number is uniformly bounded across number fields. We also generally study fields with finite p-Pythagoras number and show that the growth of the p-Pythagoras number in finite extensions is bounded.

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Section: Introductionmentioning

confidence: 96%

A p-adic analogue of Siegel's Theorem on sums of squares

Anscombe,

Dittmann,

Fehm

2019

Preprint

Self Cite

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“…Finally, in Section 7 we use our approximation theorems to discuss a related problem, namely approximation of the values of rational functions. The results of Sections 5 and 7 are crucial ingredients of our paper [ADF19b], and the results of Section 4 are used in [ADF19a].…”

Section: Introductionmentioning

confidence: 99%

Approximation theorems for spaces of localities

Anscombe,

Dittmann,

Fehm

2019

Preprint

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The classical Artin-Whaples approximation theorem allows to simultaneously approximate finitely many different elements of a field with respect to finitely many pairwise inequivalent absolute values. Several variants and generalizations exist, for example for finitely many (Krull) valuations, where one usually requires that these are independent, i.e. induce different topologies on the field. Ribenboim proved a generalization for finitely many valuations where the condition of independence is relaxed for a natural compatibility condition, and Ershov proved a statement about simultaneously approximating finitely many different elements with respect to finitely many possibly infinite sets of pairwise independent valuations.We prove approximation theorems for infinite sets of valuations and orderings without requiring pairwise independence.6 See [Pre85, p. 354], also reproven in [FHV94, Corollary 1.3]. 7 The PRC property can indeed be seen as a very strong independence assumption, since it implies in particular that distinct orderings on K induce distinct topologies, cf. [Pre85, p. 353].8 See discussion in Section 5.7 how this follows from the results in [Ers01]. 9 We use "compact" to mean what other sources call "quasi-compact", i.e. there is no implication of being a Hausdorff space.

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Denseness results in the theory of algebraic fields

Cited by 2 publications

References 10 publications

A p-adic analogue of Siegel's Theorem on sums of squares

A p-adic analogue of Siegel's Theorem on sums of squares

Approximation theorems for spaces of localities

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